Replication of Kohler et al (2016)
We can look at the groups that only contain rotations, and see if we obtain the parametric response as documented in Kohler et al. (2016).
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: rms ~ rotation + (1 | subject)
## Data: d_eeg_rot (Number of observations: 100)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Group-Level Effects:
## ~subject (Number of levels: 25)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 0.39 0.07 0.27 0.54 1.01 984 1299
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -1.15 0.11 -1.36 -0.94 1.00 1038 1757
## rotation 0.12 0.02 0.09 0.16 1.00 6242 2944
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.27 0.02 0.23 0.32 1.00 3014 3165
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
We will also investigate if we see the corresponding pattern with the display duration threshold data, with the time taken to detect the symmetry dreasing as we increase the amount of rotational symmetry.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: threshold ~ rotation + (1 | subject)
## Data: d_disp_rot (Number of observations: 46)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Group-Level Effects:
## ~subject (Number of levels: 12)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 0.48 0.17 0.18 0.87 1.00 1066 1461
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.00 0.24 -0.47 0.48 1.00 2020 2310
## rotation -0.22 0.05 -0.31 -0.12 1.00 3833 2481
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.47 0.06 0.36 0.62 1.00 2263 1904
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Index and Normality
Subgroup relations can be classified by their index, and by whether they are normal or not. Here we investigate the extent to which these two variables can account for the variation between the subgroup relationships.
First of all, we run for the eeg rms data.
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: mean_value ~ index * normal
## Data: comp_summary$eeg (Number of observations: 63)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 0.12 0.14 -0.16 0.40 1.00 2346 2695
## index 0.05 0.02 0.01 0.10 1.00 2469 2658
## normal 0.04 0.18 -0.31 0.39 1.00 1803 2367
## index:normal 0.07 0.05 -0.02 0.16 1.00 1935 2643
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.27 0.02 0.22 0.32 1.00 3103 2568
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
And now for the display duration thresholds.
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: mean_value ~ index * normal
## Data: comp_summary$threshold (Number of observations: 63)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.59 0.23 -1.05 -0.14 1.00 2382 2894
## index -0.05 0.04 -0.13 0.02 1.00 2386 2744
## normal 0.45 0.30 -0.13 1.04 1.00 2022 2321
## index:normal -0.13 0.08 -0.28 0.03 1.00 2205 2459
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.44 0.04 0.37 0.53 1.00 2634 2505
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
It is unclear from the above summary tables whether either variable has a clear effect. We can use leave-one-out metrics to compare models.
##
## Call:
## lm(formula = mean_value ~ index * normal, data = comp_summary$threshold)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.8215 -0.2812 -0.0115 0.3575 0.8305
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.59783 0.23023 -2.597 0.0119 *
## index -0.05106 0.03719 -1.373 0.1750
## normal 0.46807 0.29571 1.583 0.1188
## index:normal -0.13610 0.07604 -1.790 0.0786 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4286 on 59 degrees of freedom
## Multiple R-squared: 0.2083, Adjusted R-squared: 0.168
## F-statistic: 5.174 on 3 and 59 DF, p-value: 0.003055
##
## Call:
## lm(formula = mean_value ~ index * normal, data = comp_summary$eeg)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.83338 -0.18304 0.00269 0.17946 0.52589
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.12564 0.14074 0.893 0.3756
## index 0.05404 0.02274 2.377 0.0207 *
## normal 0.03715 0.18077 0.206 0.8379
## index:normal 0.06695 0.04648 1.440 0.1551
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.262 on 59 degrees of freedom
## Multiple R-squared: 0.2026, Adjusted R-squared: 0.1621
## F-statistic: 4.998 on 3 and 59 DF, p-value: 0.003721
We can see that the index of the subgroup relationship has an effect on both the difference in log(rms) and the difference in log(display duration): relationships with a higher index lead to larger differences.
Correlation Between Primary EEG data and Psychophysical Thresholds
Finally, we will investigate whether there is a correlation between the our primary EEG measure (rms amplitude of odd harmonics over occipital cortex) and our display duration thresholds. As our two different measures come from different samples of participants, we are unable to do a direct comparison. However, we can use the results of the models discussed in Section 3 and check for a correlation between the predicted values of the two measures.
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4370685 0.07323351 0.2785228 0.5562492
We can see that although the correlation is relatively weak, our confidence interval indicates that we can be reasonably positive that \(R^2>0\) (i.e, 95% credible interval is 0.28 - 0.56).